Your data matches 8 different statistics following compositions of up to 3 maps.
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Matching statistic: St000883
(load all 4 compositions to match this statistic)
Mapping
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,2] => 1
[2,1] => 2
[1,2,3] => 1
[1,3,2] => 2
[2,1,3] => 2
[2,3,1] => 1
[3,1,2] => 1
[3,2,1] => 3
[1,2,3,4] => 1
[1,2,4,3] => 2
[1,3,2,4] => 2
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 3
[2,1,3,4] => 2
[2,1,4,3] => 4
[2,3,1,4] => 1
[2,3,4,1] => 1
[2,4,1,3] => 3
[2,4,3,1] => 2
[3,1,2,4] => 1
[3,1,4,2] => 3
[3,2,1,4] => 3
[3,2,4,1] => 2
[3,4,1,2] => 2
[3,4,2,1] => 1
[4,1,2,3] => 1
[4,1,3,2] => 2
[4,2,1,3] => 2
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 4
[1,2,3,4,5] => 1
[1,2,3,5,4] => 2
[1,2,4,3,5] => 2
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 3
[1,3,2,4,5] => 2
[1,3,2,5,4] => 4
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 3
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 3
[1,4,3,2,5] => 3
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
Description
The number of longest increasing subsequences of a permutation.
Matching statistic: St001616
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001616: Lattices ⟶ ℤResult quality: 41% ●values known / values provided: 56%●distinct values known / distinct values provided: 41%
Values
[1] => [1] => ([],1)
 => ([],1)
 => 1
[1,2] => [2,1] => ([],2)
 => ([],1)
 => 1
[2,1] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => [3,2,1] => ([],3)
 => ([],1)
 => 1
[1,3,2] => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[2,1,3] => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => 1
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,2,3,4] => [4,3,2,1] => ([],4)
 => ([],1)
 => 1
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => 1
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([],1)
 => 1
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 1
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
 => ([],1)
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => 2
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[2,1,7,6,5,4,3] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 10
[5,4,3,2,1,7,6] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 10
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
 => ?
 => ? = 2
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => ([(0,4),(1,6),(1,7),(2,6),(2,7),(3,5),(4,3),(5,1),(5,2)],8)
 => ?
 => ? = 2
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => ([(0,5),(1,7),(2,7),(5,6),(6,1),(6,2),(7,3),(7,4)],8)
 => ?
 => ? = 2
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2),(5,7),(6,7)],8)
 => ?
 => ? = 2
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => ([(0,5),(4,3),(5,6),(6,2),(6,7),(7,1),(7,4)],8)
 => ?
 => ? = 2
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => ([(0,6),(3,7),(4,7),(5,1),(5,2),(6,3),(6,4),(7,5)],8)
 => ?
 => ? = 2
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => ([(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,1),(4,2),(4,3)],8)
 => ?
 => ? = 2
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => ([(0,3),(0,4),(3,7),(4,7),(5,6),(6,1),(6,2),(7,5)],8)
 => ?
 => ? = 2
[8,7,6,4,5,2,3,1] => [1,3,2,5,4,6,7,8] => ([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(5,7),(6,7),(7,4)],8)
 => ?
 => ? = 2
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => ([(0,4),(0,5),(0,6),(4,7),(5,7),(6,7),(7,1),(7,2),(7,3)],8)
 => ?
 => ? = 2
[7,8,3,2,4,5,6,1] => [1,6,5,4,2,3,8,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,1)],8)
 => ?
 => ? = 2
[6,5,7,4,3,2,8,1] => [1,8,2,3,4,7,5,6] => ([(0,3),(0,6),(4,7),(5,2),(6,4),(7,1),(7,5)],8)
 => ?
 => ? = 2
[3,4,5,2,6,7,8,1] => [1,8,7,6,2,5,4,3] => ([(0,4),(0,5),(0,6),(0,7),(7,1),(7,2),(7,3)],8)
 => ?
 => ? = 1
[5,2,3,4,6,7,8,1] => [1,8,7,6,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1
[3,4,2,5,6,7,8,1] => [1,8,7,6,5,2,4,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(7,1),(7,2)],8)
 => ?
 => ? = 1
[4,2,3,5,6,7,8,1] => [1,8,7,6,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
 => ?
 => ? = 1
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,2),(5,3),(7,4)],8)
 => ?
 => ? = 2
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => ([(0,6),(1,6),(3,7),(4,7),(5,3),(5,4),(6,5),(7,2)],8)
 => ?
 => ? = 2
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => ([(0,7),(1,7),(3,6),(4,6),(5,2),(6,5),(7,3),(7,4)],8)
 => ?
 => ? = 2
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,7),(6,7),(7,3)],8)
 => ?
 => ? = 2
[8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => ([(0,6),(0,7),(1,6),(1,7),(3,2),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 3
[8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
 => ?
 => ? = 2
[8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => ([(0,7),(1,7),(2,7),(4,5),(5,3),(6,4),(7,6)],8)
 => ?
 => ? = 1
[8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => ([(0,6),(1,6),(2,6),(3,7),(4,7),(5,7),(6,3),(6,4),(6,5)],8)
 => ?
 => ? = 2
[8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(5,3),(6,5),(7,5)],8)
 => ?
 => ? = 1
[8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(4,5),(6,3),(6,4),(7,3),(7,4)],8)
 => ?
 => ? = 1
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(4,3),(5,4),(6,4),(7,4)],8)
 => ?
 => ? = 2
[7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(5,3),(5,4),(6,3),(6,4),(7,3),(7,4)],8)
 => ?
 => ? = 2
[8,7,6,4,3,2,1,5] => [5,1,2,3,4,6,7,8] => ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
 => ?
 => ? = 4
[8,7,6,4,2,1,3,5] => [5,3,1,2,4,6,7,8] => ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
 => ?
 => ? = 2
[8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(7,5)],8)
 => ?
 => ? = 1
[8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ?
 => ? = 1
[7,8,5,4,2,1,3,6] => [6,3,1,2,4,5,8,7] => ([(0,7),(1,5),(1,6),(2,3),(3,7),(4,5),(4,6),(7,4)],8)
 => ?
 => ? = 2
[7,8,5,4,1,2,3,6] => [6,3,2,1,4,5,8,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(4,5),(4,6),(7,4)],8)
 => ?
 => ? = 1
[8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(7,6)],8)
 => ?
 => ? = 1
[8,6,5,4,3,1,2,7] => [7,2,1,3,4,5,6,8] => ([(0,7),(1,6),(2,6),(3,5),(4,7),(5,4),(6,3)],8)
 => ?
 => ? = 1
[8,6,5,4,2,1,3,7] => [7,3,1,2,4,5,6,8] => ([(0,7),(1,6),(2,3),(3,7),(4,5),(5,6),(7,4)],8)
 => ?
 => ? = 2
[8,6,4,3,2,1,5,7] => [7,5,1,2,3,4,6,8] => ([(0,7),(1,3),(2,6),(3,5),(4,7),(5,4),(7,6)],8)
 => ?
 => ? = 4
[8,6,4,2,1,3,5,7] => [7,5,3,1,2,4,6,8] => ([(0,6),(1,7),(2,5),(3,4),(4,7),(5,6),(7,5)],8)
 => ?
 => ? = 2
[8,2,3,4,1,5,6,7] => [7,6,5,1,4,3,2,8] => ([(0,7),(1,7),(2,7),(3,4),(3,5),(3,6),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1
[8,4,1,2,3,5,6,7] => [7,6,5,3,2,1,4,8] => ([(0,7),(1,7),(2,7),(3,6),(4,6),(5,6),(6,7)],8)
 => ?
 => ? = 1
[8,2,3,1,4,5,6,7] => [7,6,5,4,1,3,2,8] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[8,3,1,2,4,5,6,7] => [7,6,5,4,2,1,3,8] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => ?
 => ? = 1
[8,2,1,3,4,5,6,7] => [7,6,5,4,3,1,2,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
 => ?
 => ? = 2
[6,7,5,4,3,2,1,8] => [8,1,2,3,4,5,7,6] => ([(1,6),(4,5),(5,7),(6,4),(7,2),(7,3)],8)
 => ?
 => ? = 1
[6,5,7,4,3,2,1,8] => [8,1,2,3,4,7,5,6] => ([(1,6),(4,7),(5,2),(6,4),(7,3),(7,5)],8)
 => ?
 => ? = 2
[5,6,7,4,3,2,1,8] => [8,1,2,3,4,7,6,5] => ([(1,6),(5,7),(6,5),(7,2),(7,3),(7,4)],8)
 => ?
 => ? = 1
Description
The number of neutral elements in a lattice. An element $e$ of the lattice $L$ is neutral if the sublattice generated by $e$, $x$ and $y$ is distributive for all $x, y \in L$.
Matching statistic: St000550
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St000550: Lattices ⟶ ℤResult quality: 32% ●values known / values provided: 54%●distinct values known / distinct values provided: 32%
Values
[1] => [1] => ([],1)
 => ([],1)
 => 1
[1,2] => [2,1] => ([],2)
 => ([],1)
 => 1
[2,1] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => [3,2,1] => ([],3)
 => ([],1)
 => 1
[1,3,2] => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[2,1,3] => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => 1
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,2,3,4] => [4,3,2,1] => ([],4)
 => ([],1)
 => 1
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => 1
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([],1)
 => 1
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 1
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
 => ([],1)
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => 2
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[2,1,4,3,6,5] => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 8
[2,1,6,5,4,3] => [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 8
[3,2,1,6,5,4] => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 9
[3,2,6,1,5,4] => [4,5,1,6,2,3] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 8
[4,2,1,6,5,3] => [3,5,6,1,2,4] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 8
[4,3,2,1,6,5] => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 8
[1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => ([(1,6),(2,5),(3,4)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 8
[1,3,2,7,6,5,4] => [4,5,6,7,2,3,1] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 8
[1,4,3,2,7,6,5] => [5,6,7,2,3,4,1] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 9
[1,4,3,7,2,6,5] => [5,6,2,7,3,4,1] => ([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 8
[2,1,7,6,5,4,3] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 10
[5,4,3,2,1,7,6] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 10
[6,4,3,2,7,1,5] => [5,1,7,2,3,4,6] => ([(0,5),(0,6),(1,4),(1,6),(2,5),(3,2),(4,3)],7)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 8
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 8
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
 => ?
 => ? = 2
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => ([(0,4),(1,6),(1,7),(2,6),(2,7),(3,5),(4,3),(5,1),(5,2)],8)
 => ?
 => ? = 2
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => ([(0,5),(1,7),(2,7),(5,6),(6,1),(6,2),(7,3),(7,4)],8)
 => ?
 => ? = 2
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2),(5,7),(6,7)],8)
 => ?
 => ? = 2
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => ([(0,5),(4,3),(5,6),(6,2),(6,7),(7,1),(7,4)],8)
 => ?
 => ? = 2
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => ([(0,6),(3,7),(4,7),(5,1),(5,2),(6,3),(6,4),(7,5)],8)
 => ?
 => ? = 2
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => ([(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,1),(4,2),(4,3)],8)
 => ?
 => ? = 2
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => ([(0,3),(0,4),(3,7),(4,7),(5,6),(6,1),(6,2),(7,5)],8)
 => ?
 => ? = 2
[8,7,6,4,5,2,3,1] => [1,3,2,5,4,6,7,8] => ([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(5,7),(6,7),(7,4)],8)
 => ?
 => ? = 2
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => ([(0,4),(0,5),(0,6),(4,7),(5,7),(6,7),(7,1),(7,2),(7,3)],8)
 => ?
 => ? = 2
[7,8,3,2,4,5,6,1] => [1,6,5,4,2,3,8,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,1)],8)
 => ?
 => ? = 2
[6,5,7,4,3,2,8,1] => [1,8,2,3,4,7,5,6] => ([(0,3),(0,6),(4,7),(5,2),(6,4),(7,1),(7,5)],8)
 => ?
 => ? = 2
[3,4,5,2,6,7,8,1] => [1,8,7,6,2,5,4,3] => ([(0,4),(0,5),(0,6),(0,7),(7,1),(7,2),(7,3)],8)
 => ?
 => ? = 1
[5,2,3,4,6,7,8,1] => [1,8,7,6,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1
[3,4,2,5,6,7,8,1] => [1,8,7,6,5,2,4,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(7,1),(7,2)],8)
 => ?
 => ? = 1
[4,2,3,5,6,7,8,1] => [1,8,7,6,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
 => ?
 => ? = 1
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,2),(5,3),(7,4)],8)
 => ?
 => ? = 2
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => ([(0,6),(1,6),(3,7),(4,7),(5,3),(5,4),(6,5),(7,2)],8)
 => ?
 => ? = 2
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => ([(0,7),(1,7),(3,6),(4,6),(5,2),(6,5),(7,3),(7,4)],8)
 => ?
 => ? = 2
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,7),(6,7),(7,3)],8)
 => ?
 => ? = 2
[8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => ([(0,6),(0,7),(1,6),(1,7),(3,2),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 3
[8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
 => ?
 => ? = 2
[8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => ([(0,7),(1,7),(2,7),(4,5),(5,3),(6,4),(7,6)],8)
 => ?
 => ? = 1
[8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => ([(0,6),(1,6),(2,6),(3,7),(4,7),(5,7),(6,3),(6,4),(6,5)],8)
 => ?
 => ? = 2
[8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(5,3),(6,5),(7,5)],8)
 => ?
 => ? = 1
[8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(4,5),(6,3),(6,4),(7,3),(7,4)],8)
 => ?
 => ? = 1
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(4,3),(5,4),(6,4),(7,4)],8)
 => ?
 => ? = 2
[7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(5,3),(5,4),(6,3),(6,4),(7,3),(7,4)],8)
 => ?
 => ? = 2
[8,7,6,4,3,2,1,5] => [5,1,2,3,4,6,7,8] => ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
 => ?
 => ? = 4
[8,7,6,4,2,1,3,5] => [5,3,1,2,4,6,7,8] => ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
 => ?
 => ? = 2
[8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(7,5)],8)
 => ?
 => ? = 1
[8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ?
 => ? = 1
[7,8,5,4,2,1,3,6] => [6,3,1,2,4,5,8,7] => ([(0,7),(1,5),(1,6),(2,3),(3,7),(4,5),(4,6),(7,4)],8)
 => ?
 => ? = 2
[7,8,5,4,1,2,3,6] => [6,3,2,1,4,5,8,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(4,5),(4,6),(7,4)],8)
 => ?
 => ? = 1
[8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(7,6)],8)
 => ?
 => ? = 1
Description
The number of modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$, and is modular if both $(x, y)$ and $(y, x)$ are modular pairs for every $y\in L$.
Matching statistic: St000551
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St000551: Lattices ⟶ ℤResult quality: 32% ●values known / values provided: 54%●distinct values known / distinct values provided: 32%
Values
[1] => [1] => ([],1)
 => ([],1)
 => 1
[1,2] => [2,1] => ([],2)
 => ([],1)
 => 1
[2,1] => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
[1,2,3] => [3,2,1] => ([],3)
 => ([],1)
 => 1
[1,3,2] => [2,3,1] => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[2,1,3] => [3,1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 2
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => ([],1)
 => 1
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => ([],1)
 => 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(2,1)],3)
 => 3
[1,2,3,4] => [4,3,2,1] => ([],4)
 => ([],1)
 => 1
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => ([],1)
 => 1
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => ([(0,1)],2)
 => 2
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => ([],1)
 => 1
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => ([],1)
 => 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([],1)
 => 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(0,2),(2,1)],3)
 => 3
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => ([(0,1)],2)
 => 2
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => ([],1)
 => 1
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([],1)
 => 1
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,1)],2)
 => 2
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([],1)
 => 1
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([],1)
 => 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
 => ([],1)
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => ([(0,1)],2)
 => 2
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 4
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => ([],1)
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => ([],1)
 => 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => 2
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([],1)
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(0,2),(2,1)],3)
 => 3
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => ([(0,1)],2)
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2
[2,1,4,3,6,5] => [5,6,3,4,1,2] => ([(0,5),(1,4),(2,3)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 8
[2,1,6,5,4,3] => [3,4,5,6,1,2] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 8
[3,2,1,6,5,4] => [4,5,6,1,2,3] => ([(0,5),(1,4),(4,2),(5,3)],6)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 9
[3,2,6,1,5,4] => [4,5,1,6,2,3] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 8
[4,2,1,6,5,3] => [3,5,6,1,2,4] => ([(0,3),(1,4),(1,5),(3,5),(4,2)],6)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 8
[4,3,2,1,6,5] => [5,6,1,2,3,4] => ([(0,5),(1,3),(4,2),(5,4)],6)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 8
[1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => ([(1,6),(2,5),(3,4)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => ? = 8
[1,3,2,7,6,5,4] => [4,5,6,7,2,3,1] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 8
[1,4,3,2,7,6,5] => [5,6,7,2,3,4,1] => ([(1,6),(2,5),(5,3),(6,4)],7)
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 9
[1,4,3,7,2,6,5] => [5,6,2,7,3,4,1] => ([(1,4),(2,3),(2,6),(3,5),(4,6)],7)
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 8
[2,1,7,6,5,4,3] => [3,4,5,6,7,1,2] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 10
[5,4,3,2,1,7,6] => [6,7,1,2,3,4,5] => ([(0,6),(1,3),(4,5),(5,2),(6,4)],7)
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 10
[6,4,3,2,7,1,5] => [5,1,7,2,3,4,6] => ([(0,5),(0,6),(1,4),(1,6),(2,5),(3,2),(4,3)],7)
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 8
[8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ([(0,7),(2,4),(3,2),(4,6),(5,3),(6,1),(7,5)],8)
 => ? = 8
[7,6,8,5,4,3,2,1] => [1,2,3,4,5,8,6,7] => ([(0,6),(3,4),(4,7),(5,1),(6,3),(7,2),(7,5)],8)
 => ?
 => ? = 2
[7,8,5,6,4,3,2,1] => [1,2,3,4,6,5,8,7] => ([(0,4),(1,6),(1,7),(2,6),(2,7),(3,5),(4,3),(5,1),(5,2)],8)
 => ?
 => ? = 2
[7,8,6,4,5,3,2,1] => [1,2,3,5,4,6,8,7] => ([(0,5),(1,7),(2,7),(5,6),(6,1),(6,2),(7,3),(7,4)],8)
 => ?
 => ? = 2
[8,6,7,4,5,3,2,1] => [1,2,3,5,4,7,6,8] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2),(5,7),(6,7)],8)
 => ?
 => ? = 2
[6,5,7,4,8,3,2,1] => [1,2,3,8,4,7,5,6] => ([(0,5),(4,3),(5,6),(6,2),(6,7),(7,1),(7,4)],8)
 => ?
 => ? = 2
[7,8,6,5,3,4,2,1] => [1,2,4,3,5,6,8,7] => ([(0,6),(3,7),(4,7),(5,1),(5,2),(6,3),(6,4),(7,5)],8)
 => ?
 => ? = 2
[6,7,8,3,4,5,2,1] => [1,2,5,4,3,8,7,6] => ([(0,4),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,1),(4,2),(4,3)],8)
 => ?
 => ? = 2
[7,8,6,5,4,2,3,1] => [1,3,2,4,5,6,8,7] => ([(0,3),(0,4),(3,7),(4,7),(5,6),(6,1),(6,2),(7,5)],8)
 => ?
 => ? = 2
[8,7,6,4,5,2,3,1] => [1,3,2,5,4,6,7,8] => ([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(5,7),(6,7),(7,4)],8)
 => ?
 => ? = 2
[6,7,8,5,2,3,4,1] => [1,4,3,2,5,8,7,6] => ([(0,4),(0,5),(0,6),(4,7),(5,7),(6,7),(7,1),(7,2),(7,3)],8)
 => ?
 => ? = 2
[7,8,3,2,4,5,6,1] => [1,6,5,4,2,3,8,7] => ([(0,2),(0,3),(0,4),(0,5),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,1)],8)
 => ?
 => ? = 2
[6,5,7,4,3,2,8,1] => [1,8,2,3,4,7,5,6] => ([(0,3),(0,6),(4,7),(5,2),(6,4),(7,1),(7,5)],8)
 => ?
 => ? = 2
[3,4,5,2,6,7,8,1] => [1,8,7,6,2,5,4,3] => ([(0,4),(0,5),(0,6),(0,7),(7,1),(7,2),(7,3)],8)
 => ?
 => ? = 1
[5,2,3,4,6,7,8,1] => [1,8,7,6,4,3,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(4,7),(5,7),(6,7)],8)
 => ?
 => ? = 1
[3,4,2,5,6,7,8,1] => [1,8,7,6,5,2,4,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(7,1),(7,2)],8)
 => ?
 => ? = 1
[4,2,3,5,6,7,8,1] => [1,8,7,6,5,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(5,7),(6,7)],8)
 => ?
 => ? = 1
[8,7,6,5,4,3,1,2] => [2,1,3,4,5,6,7,8] => ([(0,7),(1,7),(3,5),(4,3),(5,2),(6,4),(7,6)],8)
 => ?
 => ? = 1
[8,6,7,5,4,3,1,2] => [2,1,3,4,5,7,6,8] => ([(0,7),(1,7),(2,6),(3,6),(4,5),(5,2),(5,3),(7,4)],8)
 => ?
 => ? = 2
[8,7,5,6,4,3,1,2] => [2,1,3,4,6,5,7,8] => ([(0,6),(1,6),(3,7),(4,7),(5,3),(5,4),(6,5),(7,2)],8)
 => ?
 => ? = 2
[8,7,6,4,5,3,1,2] => [2,1,3,5,4,6,7,8] => ([(0,7),(1,7),(3,6),(4,6),(5,2),(6,5),(7,3),(7,4)],8)
 => ?
 => ? = 2
[8,7,6,5,3,4,1,2] => [2,1,4,3,5,6,7,8] => ([(0,5),(0,6),(1,5),(1,6),(3,4),(4,2),(5,7),(6,7),(7,3)],8)
 => ?
 => ? = 2
[8,7,5,6,3,4,1,2] => [2,1,4,3,6,5,7,8] => ([(0,6),(0,7),(1,6),(1,7),(3,2),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ?
 => ? = 3
[8,7,6,5,4,2,1,3] => [3,1,2,4,5,6,7,8] => ([(0,7),(1,3),(3,7),(4,5),(5,2),(6,4),(7,6)],8)
 => ?
 => ? = 2
[8,7,6,5,4,1,2,3] => [3,2,1,4,5,6,7,8] => ([(0,7),(1,7),(2,7),(4,5),(5,3),(6,4),(7,6)],8)
 => ?
 => ? = 1
[8,5,6,7,4,1,2,3] => [3,2,1,4,7,6,5,8] => ([(0,6),(1,6),(2,6),(3,7),(4,7),(5,7),(6,3),(6,4),(6,5)],8)
 => ?
 => ? = 2
[8,7,6,4,5,1,2,3] => [3,2,1,5,4,6,7,8] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,4),(5,3),(6,5),(7,5)],8)
 => ?
 => ? = 1
[8,6,7,4,5,1,2,3] => [3,2,1,5,4,7,6,8] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,5),(4,5),(6,3),(6,4),(7,3),(7,4)],8)
 => ?
 => ? = 1
[8,7,4,5,6,1,2,3] => [3,2,1,6,5,4,7,8] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(4,3),(5,4),(6,4),(7,4)],8)
 => ?
 => ? = 2
[7,8,4,5,6,1,2,3] => [3,2,1,6,5,4,8,7] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(5,3),(5,4),(6,3),(6,4),(7,3),(7,4)],8)
 => ?
 => ? = 2
[8,7,6,4,3,2,1,5] => [5,1,2,3,4,6,7,8] => ([(0,7),(1,5),(3,7),(4,3),(5,4),(6,2),(7,6)],8)
 => ?
 => ? = 4
[8,7,6,4,2,1,3,5] => [5,3,1,2,4,6,7,8] => ([(0,7),(1,6),(2,3),(3,7),(4,5),(6,4),(7,6)],8)
 => ?
 => ? = 2
[8,7,6,1,2,3,4,5] => [5,4,3,2,1,6,7,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(7,5)],8)
 => ?
 => ? = 1
[8,6,7,1,2,3,4,5] => [5,4,3,2,1,7,6,8] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
 => ?
 => ? = 1
[7,8,5,4,2,1,3,6] => [6,3,1,2,4,5,8,7] => ([(0,7),(1,5),(1,6),(2,3),(3,7),(4,5),(4,6),(7,4)],8)
 => ?
 => ? = 2
[7,8,5,4,1,2,3,6] => [6,3,2,1,4,5,8,7] => ([(0,7),(1,7),(2,7),(3,5),(3,6),(4,5),(4,6),(7,4)],8)
 => ?
 => ? = 1
[8,7,1,2,3,4,5,6] => [6,5,4,3,2,1,7,8] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(7,6)],8)
 => ?
 => ? = 1
Description
The number of left modular elements of a lattice. A pair $(x, y)$ of elements of a lattice $L$ is a modular pair if for every $z\geq y$ we have that $(y\vee x) \wedge z = y \vee (x \wedge z)$. An element $x$ is left-modular if $(x, y)$ is a modular pair for every $y\in L$.
Matching statistic: St000363
(load all 2 compositions to match this statistic)
Mapping
Mp00160: Permutations —graph of inversions⟶ Graphs
St000363: Graphs ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 45%
Values
[1] => ([],1)
 => 1
[1,2] => ([],2)
 => 1
[2,1] => ([(0,1)],2)
 => 2
[1,2,3] => ([],3)
 => 1
[1,3,2] => ([(1,2)],3)
 => 2
[2,1,3] => ([(1,2)],3)
 => 2
[2,3,1] => ([(0,2),(1,2)],3)
 => 1
[3,1,2] => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 3
[1,2,3,4] => ([],4)
 => 1
[1,2,4,3] => ([(2,3)],4)
 => 2
[1,3,2,4] => ([(2,3)],4)
 => 2
[1,3,4,2] => ([(1,3),(2,3)],4)
 => 1
[1,4,2,3] => ([(1,3),(2,3)],4)
 => 1
[1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
[2,1,3,4] => ([(2,3)],4)
 => 2
[2,1,4,3] => ([(0,3),(1,2)],4)
 => 4
[2,3,1,4] => ([(1,3),(2,3)],4)
 => 1
[2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 3
[2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,1,2,4] => ([(1,3),(2,3)],4)
 => 1
[3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 3
[3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4
[1,2,3,4,5] => ([],5)
 => 1
[1,2,3,5,4] => ([(3,4)],5)
 => 2
[1,2,4,3,5] => ([(3,4)],5)
 => 2
[1,2,4,5,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,3,4] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,3,2,4,5] => ([(3,4)],5)
 => 2
[1,3,2,5,4] => ([(1,4),(2,3)],5)
 => 4
[1,3,4,2,5] => ([(2,4),(3,4)],5)
 => 1
[1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,3,5,4,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,3,5] => ([(2,4),(3,4)],5)
 => 1
[1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[8,7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 8
[7,6,8,5,4,3,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,8,5,6,4,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,8,6,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,6,7,4,5,3,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,6,5,4,8,3,2,1] => ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,5,7,4,8,3,2,1] => ([(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,8,6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,6,7,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,7,5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[6,7,8,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,8,6,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,6,7,5,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,7,5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,7,6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,6,5,7,3,2,4,1] => ([(0,1),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[6,7,8,5,2,3,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,5,6,7,2,3,4,1] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,8,3,2,4,5,6,1] => ([(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 2
[7,6,5,4,3,2,8,1] => ([(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 6
[6,5,7,4,3,2,8,1] => ([(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[6,5,4,3,7,2,8,1] => ([(0,7),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[5,4,6,3,7,2,8,1] => ([(0,7),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[3,4,5,2,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[5,2,3,4,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,3,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[3,4,2,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,2,3,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[3,2,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => ? = 1
[8,7,6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[7,8,6,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,6,7,5,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,7,5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,7,6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,7,6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,7,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 3
[7,8,5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 4
[5,6,7,8,3,4,1,2] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1
[7,8,3,4,5,6,1,2] => ([(0,4),(0,5),(0,6),(0,7),(1,4),(1,5),(1,6),(1,7),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1
[3,4,5,6,7,8,1,2] => ([(0,6),(0,7),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 1
[8,7,6,5,4,2,1,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,6,8,5,4,2,1,3] => ([(0,1),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 4
[8,7,6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[6,7,8,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,5,6,7,4,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[8,7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[8,6,7,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,7),(6,7)],8)
 => ? = 1
[8,7,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
[7,8,4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,3),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
 => ? = 2
Description
The number of minimal vertex covers of a graph. A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. A vertex cover is minimal if it contains the least possible number of vertices. This is also the leading coefficient of the clique polynomial of the complement of $G$. This is also the number of independent sets of maximal cardinality of $G$.
Matching statistic: St000909
(load all 2 compositions to match this statistic)
Mapping
Mp00065: Permutations —permutation poset⟶ Posets
St000909: Posets ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 59%
Values
[1] => ([],1)
 => 1
[1,2] => ([(0,1)],2)
 => 1
[2,1] => ([],2)
 => 2
[1,2,3] => ([(0,2),(2,1)],3)
 => 1
[1,3,2] => ([(0,1),(0,2)],3)
 => 2
[2,1,3] => ([(0,2),(1,2)],3)
 => 2
[2,3,1] => ([(1,2)],3)
 => 1
[3,1,2] => ([(1,2)],3)
 => 1
[3,2,1] => ([],3)
 => 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 1
[1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 1
[1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 3
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 4
[2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 1
[2,3,4,1] => ([(1,2),(2,3)],4)
 => 1
[2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 3
[2,4,3,1] => ([(1,2),(1,3)],4)
 => 2
[3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 1
[3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 3
[3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 3
[3,2,4,1] => ([(1,3),(2,3)],4)
 => 2
[3,4,1,2] => ([(0,3),(1,2)],4)
 => 2
[3,4,2,1] => ([(2,3)],4)
 => 1
[4,1,2,3] => ([(1,2),(2,3)],4)
 => 1
[4,1,3,2] => ([(1,2),(1,3)],4)
 => 2
[4,2,1,3] => ([(1,3),(2,3)],4)
 => 2
[4,2,3,1] => ([(2,3)],4)
 => 1
[4,3,1,2] => ([(2,3)],4)
 => 1
[4,3,2,1] => ([],4)
 => 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1
[1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
 => 2
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 2
[1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 1
[1,2,5,3,4] => ([(0,4),(3,2),(4,1),(4,3)],5)
 => 1
[1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5)
 => 3
[1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 2
[1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
 => 4
[1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 1
[1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
 => 1
[1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 3
[1,3,5,4,2] => ([(0,3),(0,4),(4,1),(4,2)],5)
 => 2
[1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
 => 1
[1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
 => 3
[1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 3
[1,4,3,5,2] => ([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
 => 2
[1,2,3,5,4,7,6] => ([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
 => ? = 4
[1,2,3,5,7,4,6] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => ? = 3
[1,2,3,6,4,7,5] => ([(0,4),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3)],7)
 => ? = 3
[1,2,4,3,5,7,6] => ([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
 => ? = 4
[1,2,4,3,6,5,7] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
 => ? = 4
[1,2,4,3,6,7,5] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ? = 2
[1,2,4,3,7,5,6] => ([(0,4),(2,5),(2,6),(3,5),(3,6),(4,2),(4,3),(6,1)],7)
 => ? = 2
[1,2,4,3,7,6,5] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,1),(3,2)],7)
 => ? = 6
[1,2,4,5,3,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
 => ? = 2
[1,2,4,5,7,3,6] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
 => ? = 2
[1,2,4,6,3,7,5] => ([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
 => ? = 1
[1,2,4,6,5,7,3] => ([(0,5),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ? = 2
[1,2,4,6,7,3,5] => ([(0,5),(2,6),(3,1),(4,3),(4,6),(5,2),(5,4)],7)
 => ? = 1
[1,2,4,7,3,5,6] => ([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
 => ? = 2
[1,2,4,7,3,6,5] => ([(0,4),(2,5),(2,6),(3,1),(3,5),(3,6),(4,2),(4,3)],7)
 => ? = 5
[1,2,4,7,5,3,6] => ([(0,5),(2,6),(3,6),(4,1),(4,3),(5,2),(5,4)],7)
 => ? = 1
[1,2,4,7,6,3,5] => ([(0,5),(3,6),(4,1),(4,2),(4,6),(5,3),(5,4)],7)
 => ? = 4
[1,2,5,3,4,7,6] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,2),(4,1),(4,3)],7)
 => ? = 2
[1,2,5,3,6,7,4] => ([(0,5),(2,6),(4,1),(4,6),(5,2),(5,4),(6,3)],7)
 => ? = 2
[1,2,5,3,7,4,6] => ([(0,4),(1,5),(2,5),(2,6),(3,1),(3,6),(4,2),(4,3)],7)
 => ? = 1
[1,2,5,3,7,6,4] => ([(0,4),(2,5),(2,6),(3,1),(3,5),(3,6),(4,2),(4,3)],7)
 => ? = 5
[1,2,5,4,6,7,3] => ([(0,5),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ? = 2
[1,2,5,6,3,7,4] => ([(0,5),(2,6),(3,2),(4,1),(4,6),(5,3),(5,4)],7)
 => ? = 1
[1,2,5,6,4,7,3] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ? = 1
[1,2,5,7,3,4,6] => ([(0,5),(2,6),(3,2),(4,1),(4,6),(5,3),(5,4)],7)
 => ? = 1
[1,2,5,7,3,6,4] => ([(0,5),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4)],7)
 => ? = 4
[1,2,5,7,4,3,6] => ([(0,5),(2,6),(3,6),(4,1),(4,6),(5,2),(5,3),(5,4)],7)
 => ? = 4
[1,2,5,7,4,6,3] => ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
 => ? = 3
[1,2,6,3,4,7,5] => ([(0,5),(2,6),(3,4),(4,1),(4,6),(5,2),(5,3)],7)
 => ? = 2
[1,2,6,3,5,7,4] => ([(0,5),(2,6),(3,6),(4,1),(4,3),(5,2),(5,4)],7)
 => ? = 1
[1,2,6,3,7,4,5] => ([(0,5),(2,6),(3,1),(4,3),(4,6),(5,2),(5,4)],7)
 => ? = 1
[1,2,6,3,7,5,4] => ([(0,5),(3,6),(4,1),(4,2),(4,6),(5,3),(5,4)],7)
 => ? = 4
[1,2,6,4,5,7,3] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ? = 1
[1,2,6,4,7,3,5] => ([(0,4),(1,6),(2,5),(3,5),(3,6),(4,1),(4,2),(4,3)],7)
 => ? = 4
[1,2,6,4,7,5,3] => ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
 => ? = 3
[1,2,6,5,3,7,4] => ([(0,5),(2,6),(3,6),(4,1),(4,6),(5,2),(5,3),(5,4)],7)
 => ? = 4
[1,2,6,5,7,3,4] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(5,4)],7)
 => ? = 3
[1,2,6,7,4,3,5] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(5,4)],7)
 => ? = 3
[1,2,7,3,5,4,6] => ([(0,5),(2,6),(3,6),(4,2),(4,3),(5,1),(5,4)],7)
 => ? = 2
[1,2,7,4,3,5,6] => ([(0,5),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
 => ? = 2
[1,2,7,4,5,3,6] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ? = 1
[1,2,7,4,6,3,5] => ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
 => ? = 3
[1,2,7,5,3,4,6] => ([(0,5),(2,6),(3,6),(4,3),(5,1),(5,2),(5,4)],7)
 => ? = 1
[1,2,7,5,3,6,4] => ([(0,5),(3,6),(4,2),(4,6),(5,1),(5,3),(5,4)],7)
 => ? = 3
[1,3,2,4,5,7,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
 => ? = 4
[1,3,2,4,6,7,5] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(6,2),(6,5)],7)
 => ? = 2
[1,3,2,4,7,5,6] => ([(0,3),(0,4),(3,6),(4,6),(5,1),(6,2),(6,5)],7)
 => ? = 2
[1,3,2,5,6,4,7] => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ? = 2
[1,3,2,5,6,7,4] => ([(0,2),(0,3),(2,5),(2,6),(3,5),(3,6),(4,1),(6,4)],7)
 => ? = 2
[1,3,2,6,4,5,7] => ([(0,2),(0,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,5),(6,1)],7)
 => ? = 2
Description
The number of maximal chains of maximal size in a poset.
Matching statistic: St000911
(load all 2 compositions to match this statistic)
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000911: Posets ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 59%
Values
[1] => [1] => ([],1)
 => 1
[1,2] => [2,1] => ([],2)
 => 1
[2,1] => [1,2] => ([(0,1)],2)
 => 2
[1,2,3] => [3,2,1] => ([],3)
 => 1
[1,3,2] => [2,3,1] => ([(1,2)],3)
 => 2
[2,1,3] => [3,1,2] => ([(1,2)],3)
 => 2
[2,3,1] => [1,3,2] => ([(0,1),(0,2)],3)
 => 1
[3,1,2] => [2,1,3] => ([(0,2),(1,2)],3)
 => 1
[3,2,1] => [1,2,3] => ([(0,2),(2,1)],3)
 => 3
[1,2,3,4] => [4,3,2,1] => ([],4)
 => 1
[1,2,4,3] => [3,4,2,1] => ([(2,3)],4)
 => 2
[1,3,2,4] => [4,2,3,1] => ([(2,3)],4)
 => 2
[1,3,4,2] => [2,4,3,1] => ([(1,2),(1,3)],4)
 => 1
[1,4,2,3] => [3,2,4,1] => ([(1,3),(2,3)],4)
 => 1
[1,4,3,2] => [2,3,4,1] => ([(1,2),(2,3)],4)
 => 3
[2,1,3,4] => [4,3,1,2] => ([(2,3)],4)
 => 2
[2,1,4,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
 => 4
[2,3,1,4] => [4,1,3,2] => ([(1,2),(1,3)],4)
 => 1
[2,3,4,1] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
 => 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
 => 3
[2,4,3,1] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
 => 2
[3,1,2,4] => [4,2,1,3] => ([(1,3),(2,3)],4)
 => 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
 => 3
[3,2,1,4] => [4,1,2,3] => ([(1,2),(2,3)],4)
 => 3
[3,2,4,1] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
 => 2
[3,4,1,2] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2
[3,4,2,1] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
 => 1
[4,1,2,3] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,3,2] => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[4,2,1,3] => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => 2
[4,2,3,1] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => 1
[4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => ([],5)
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => ([(3,4)],5)
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => ([(3,4)],5)
 => 2
[1,2,4,5,3] => [3,5,4,2,1] => ([(2,3),(2,4)],5)
 => 1
[1,2,5,3,4] => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => ([(3,4)],5)
 => 2
[1,3,2,5,4] => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => 4
[1,3,4,2,5] => [5,2,4,3,1] => ([(2,3),(2,4)],5)
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
 => 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(1,4),(2,3),(2,4)],5)
 => 3
[1,3,5,4,2] => [2,4,5,3,1] => ([(1,3),(1,4),(4,2)],5)
 => 2
[1,4,2,3,5] => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => ([(1,4),(2,3),(2,4)],5)
 => 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => 3
[1,4,3,5,2] => [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => 2
[1,2,3,7,4,6,5] => [5,6,4,7,3,2,1] => ([(3,6),(4,5),(5,6)],7)
 => ? = 2
[1,2,3,7,5,4,6] => [6,4,5,7,3,2,1] => ([(3,6),(4,5),(5,6)],7)
 => ? = 2
[1,2,4,3,7,6,5] => [5,6,7,3,4,2,1] => ([(2,4),(3,5),(5,6)],7)
 => ? = 6
[1,2,4,6,3,7,5] => [5,7,3,6,4,2,1] => ([(2,5),(2,6),(3,4),(3,6)],7)
 => ? = 1
[1,2,4,6,7,3,5] => [5,3,7,6,4,2,1] => ([(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 1
[1,2,4,7,3,6,5] => [5,6,3,7,4,2,1] => ([(2,5),(3,4),(3,6),(5,6)],7)
 => ? = 5
[1,2,4,7,5,3,6] => [6,3,5,7,4,2,1] => ([(2,6),(3,4),(3,5),(5,6)],7)
 => ? = 1
[1,2,4,7,6,5,3] => [3,5,6,7,4,2,1] => ([(2,4),(2,6),(5,3),(6,5)],7)
 => ? = 3
[1,2,5,4,3,7,6] => [6,7,3,4,5,2,1] => ([(2,4),(3,5),(5,6)],7)
 => ? = 6
[1,2,5,6,3,7,4] => [4,7,3,6,5,2,1] => ([(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 1
[1,2,5,6,7,3,4] => [4,3,7,6,5,2,1] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
 => ? = 1
[1,2,5,7,3,6,4] => [4,6,3,7,5,2,1] => ([(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ? = 4
[1,2,5,7,4,3,6] => [6,3,4,7,5,2,1] => ([(2,6),(3,4),(4,5),(4,6)],7)
 => ? = 4
[1,2,5,7,6,4,3] => [3,4,6,7,5,2,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
 => ? = 2
[1,2,6,3,5,4,7] => [7,4,5,3,6,2,1] => ([(3,6),(4,5),(5,6)],7)
 => ? = 2
[1,2,6,3,5,7,4] => [4,7,5,3,6,2,1] => ([(2,6),(3,4),(3,5),(5,6)],7)
 => ? = 1
[1,2,6,3,7,5,4] => [4,5,7,3,6,2,1] => ([(2,6),(3,4),(4,5),(4,6)],7)
 => ? = 4
[1,2,6,4,3,5,7] => [7,5,3,4,6,2,1] => ([(3,6),(4,5),(5,6)],7)
 => ? = 2
[1,2,6,4,3,7,5] => [5,7,3,4,6,2,1] => ([(2,5),(3,4),(3,6),(5,6)],7)
 => ? = 5
[1,2,6,4,7,3,5] => [5,3,7,4,6,2,1] => ([(2,5),(2,6),(3,4),(3,5),(4,6)],7)
 => ? = 4
[1,2,6,5,4,7,3] => [3,7,4,5,6,2,1] => ([(2,4),(2,6),(5,3),(6,5)],7)
 => ? = 3
[1,2,6,5,7,4,3] => [3,4,7,5,6,2,1] => ([(2,6),(5,4),(6,3),(6,5)],7)
 => ? = 2
[1,2,6,7,3,5,4] => [4,5,3,7,6,2,1] => ([(2,5),(2,6),(3,4),(4,5),(4,6)],7)
 => ? = 3
[1,2,6,7,4,3,5] => [5,3,4,7,6,2,1] => ([(2,5),(2,6),(3,4),(4,5),(4,6)],7)
 => ? = 3
[1,2,7,3,5,6,4] => [4,6,5,3,7,2,1] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1
[1,2,7,3,6,5,4] => [4,5,6,3,7,2,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 3
[1,2,7,4,3,6,5] => [5,6,3,4,7,2,1] => ([(2,5),(3,4),(4,6),(5,6)],7)
 => ? = 4
[1,2,7,4,5,3,6] => [6,3,5,4,7,2,1] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1
[1,2,7,5,4,3,6] => [6,3,4,5,7,2,1] => ([(2,6),(3,4),(4,5),(5,6)],7)
 => ? = 3
[1,2,7,5,6,4,3] => [3,4,6,5,7,2,1] => ([(2,3),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 1
[1,2,7,6,3,5,4] => [4,5,3,6,7,2,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
 => ? = 2
[1,2,7,6,4,3,5] => [5,3,4,6,7,2,1] => ([(2,6),(3,4),(4,6),(6,5)],7)
 => ? = 2
[1,2,7,6,5,4,3] => [3,4,5,6,7,2,1] => ([(2,6),(4,5),(5,3),(6,4)],7)
 => ? = 5
[1,3,2,4,7,6,5] => [5,6,7,4,2,3,1] => ([(2,4),(3,5),(5,6)],7)
 => ? = 6
[1,3,2,5,4,7,6] => [6,7,4,5,2,3,1] => ([(1,6),(2,5),(3,4)],7)
 => ? = 8
[1,3,2,5,7,4,6] => [6,4,7,5,2,3,1] => ([(1,6),(2,5),(3,4),(3,6)],7)
 => ? = 6
[1,3,2,5,7,6,4] => [4,6,7,5,2,3,1] => ([(1,5),(2,4),(2,6),(6,3)],7)
 => ? = 4
[1,3,2,6,4,7,5] => [5,7,4,6,2,3,1] => ([(1,6),(2,5),(3,4),(3,6)],7)
 => ? = 6
[1,3,2,6,5,4,7] => [7,4,5,6,2,3,1] => ([(2,4),(3,5),(5,6)],7)
 => ? = 6
[1,3,2,6,5,7,4] => [4,7,5,6,2,3,1] => ([(1,5),(2,4),(2,6),(6,3)],7)
 => ? = 4
[1,3,2,6,7,5,4] => [4,5,7,6,2,3,1] => ([(1,5),(2,6),(6,3),(6,4)],7)
 => ? = 2
[1,3,2,7,4,6,5] => [5,6,4,7,2,3,1] => ([(1,6),(2,4),(3,5),(5,6)],7)
 => ? = 4
[1,3,2,7,5,4,6] => [6,4,5,7,2,3,1] => ([(1,6),(2,4),(3,5),(5,6)],7)
 => ? = 4
[1,3,2,7,5,6,4] => [4,6,5,7,2,3,1] => ([(1,3),(2,4),(2,5),(4,6),(5,6)],7)
 => ? = 2
[1,3,2,7,6,5,4] => [4,5,6,7,2,3,1] => ([(1,6),(2,4),(5,3),(6,5)],7)
 => ? = 8
[1,3,4,2,6,7,5] => [5,7,6,2,4,3,1] => ([(1,5),(1,6),(2,3),(2,4)],7)
 => ? = 1
[1,3,4,2,7,6,5] => [5,6,7,2,4,3,1] => ([(1,6),(2,4),(2,5),(6,3)],7)
 => ? = 3
[1,3,4,6,2,7,5] => [5,7,2,6,4,3,1] => ([(1,5),(1,6),(2,3),(2,4),(2,6)],7)
 => ? = 1
[1,3,4,6,7,2,5] => [5,2,7,6,4,3,1] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6)],7)
 => ? = 1
[1,3,4,7,2,6,5] => [5,6,2,7,4,3,1] => ([(1,5),(2,3),(2,4),(2,6),(5,6)],7)
 => ? = 3
Description
The number of maximal antichains of maximal size in a poset.
Matching statistic: St000993
Mapping
Mp00064: Permutations —reverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00251: Graphs —clique sizes⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 31% ●values known / values provided: 31%●distinct values known / distinct values provided: 32%
Values
[1] => [1] => ([],1)
 => [1]
 => ? = 1
[1,2] => [2,1] => ([(0,1)],2)
 => [2]
 => 1
[2,1] => [1,2] => ([],2)
 => [1,1]
 => 2
[1,2,3] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => [3]
 => 1
[1,3,2] => [2,3,1] => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
[2,1,3] => [3,1,2] => ([(0,2),(1,2)],3)
 => [2,2]
 => 2
[2,3,1] => [1,3,2] => ([(1,2)],3)
 => [2,1]
 => 1
[3,1,2] => [2,1,3] => ([(1,2)],3)
 => [2,1]
 => 1
[3,2,1] => [1,2,3] => ([],3)
 => [1,1,1]
 => 3
[1,2,3,4] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [4]
 => 1
[1,2,4,3] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 2
[1,3,2,4] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 2
[1,3,4,2] => [2,4,3,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 1
[1,4,2,3] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 1
[1,4,3,2] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
[2,1,3,4] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => [3,3]
 => 2
[2,1,4,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => [2,2,2,2]
 => 4
[2,3,1,4] => [4,1,3,2] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 1
[2,3,4,1] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[2,4,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
[2,4,3,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
[3,1,2,4] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => [3,2]
 => 1
[3,1,4,2] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => [2,2,2]
 => 3
[3,2,1,4] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => [2,2,2]
 => 3
[3,2,4,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
[3,4,1,2] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => [2,2]
 => 2
[3,4,2,1] => [1,2,4,3] => ([(2,3)],4)
 => [2,1,1]
 => 1
[4,1,2,3] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
 => [3,1]
 => 1
[4,1,3,2] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
[4,2,1,3] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => [2,2,1]
 => 2
[4,2,3,1] => [1,3,2,4] => ([(2,3)],4)
 => [2,1,1]
 => 1
[4,3,1,2] => [2,1,3,4] => ([(2,3)],4)
 => [2,1,1]
 => 1
[4,3,2,1] => [1,2,3,4] => ([],4)
 => [1,1,1,1]
 => 4
[1,2,3,4,5] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [5]
 => 1
[1,2,3,5,4] => [4,5,3,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => 2
[1,2,4,3,5] => [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => 2
[1,2,4,5,3] => [3,5,4,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => 1
[1,2,5,3,4] => [4,3,5,2,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => 1
[1,2,5,4,3] => [3,4,5,2,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 3
[1,3,2,4,5] => [5,4,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,4]
 => 2
[1,3,2,5,4] => [4,5,2,3,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => [3,3,3,3]
 => 4
[1,3,4,2,5] => [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => 1
[1,3,4,5,2] => [2,5,4,3,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,2]
 => 1
[1,3,5,2,4] => [4,2,5,3,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 3
[1,3,5,4,2] => [2,4,5,3,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => 2
[1,4,2,3,5] => [5,3,2,4,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [4,3]
 => 1
[1,4,2,5,3] => [3,5,2,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 3
[1,4,3,2,5] => [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,3]
 => 3
[1,4,3,5,2] => [2,5,3,4,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,3,2]
 => 2
[1,4,5,2,3] => [3,2,5,4,1] => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => [3,3]
 => 2
[1,4,5,3,2] => [2,3,5,4,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => [3,2,2]
 => 1
[1,2,4,3,6,5] => [5,6,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,4,4,4]
 => ? = 4
[1,3,2,4,6,5] => [5,6,4,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,4,4,4]
 => ? = 4
[1,3,2,5,4,6] => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,4,4,4]
 => ? = 4
[1,3,2,5,6,4] => [4,6,5,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => ? = 2
[1,3,2,6,4,5] => [5,4,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => ? = 2
[1,3,2,6,5,4] => [4,5,6,2,3,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,3,3]
 => ? = 6
[1,3,4,2,6,5] => [5,6,2,4,3,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => ? = 2
[1,3,5,2,6,4] => [4,6,2,5,3,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,3,3,3]
 => ? = 1
[1,3,6,2,5,4] => [4,5,2,6,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => ? = 5
[1,4,2,3,6,5] => [5,6,3,2,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => ? = 2
[1,4,2,6,3,5] => [5,3,6,2,4,1] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,3,3,3]
 => ? = 1
[1,4,2,6,5,3] => [3,5,6,2,4,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => ? = 5
[1,4,3,2,6,5] => [5,6,2,3,4,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,3,3]
 => ? = 6
[1,4,3,6,2,5] => [5,2,6,3,4,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => ? = 5
[1,4,3,6,5,2] => [2,5,6,3,4,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => ? = 4
[1,5,3,2,6,4] => [4,6,2,3,5,1] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => ? = 5
[1,6,3,2,5,4] => [4,5,2,3,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => ? = 4
[2,1,3,4,6,5] => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,4,4,4]
 => ? = 4
[2,1,3,5,4,6] => [6,4,5,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,4,4,4]
 => ? = 4
[2,1,3,5,6,4] => [4,6,5,3,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => ? = 2
[2,1,3,6,4,5] => [5,4,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => ? = 2
[2,1,3,6,5,4] => [4,5,6,3,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,3,3]
 => ? = 6
[2,1,4,3,5,6] => [6,5,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,4,4,4]
 => ? = 4
[2,1,4,3,6,5] => [5,6,3,4,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,3,3,3,3,3,3]
 => ? = 8
[2,1,4,5,3,6] => [6,3,5,4,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => ? = 2
[2,1,4,6,3,5] => [5,3,6,4,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,3,3,3,3]
 => ? = 6
[2,1,4,6,5,3] => [3,5,6,4,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,3,3,2,2]
 => ? = 4
[2,1,5,3,4,6] => [6,4,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => ? = 2
[2,1,5,3,6,4] => [4,6,3,5,1,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,3,3,3,3]
 => ? = 6
[2,1,5,4,3,6] => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,3,3]
 => ? = 6
[2,1,5,4,6,3] => [3,6,4,5,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,3,3,2,2]
 => ? = 4
[2,1,5,6,4,3] => [3,4,6,5,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => ? = 2
[2,1,6,3,5,4] => [4,5,3,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,3,3,2,2]
 => ? = 4
[2,1,6,4,3,5] => [5,3,4,6,1,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,3,3,2,2]
 => ? = 4
[2,1,6,4,5,3] => [3,5,4,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => ? = 2
[2,1,6,5,3,4] => [4,3,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => ? = 2
[2,1,6,5,4,3] => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [2,2,2,2,2,2,2,2]
 => ? = 8
[2,3,1,4,6,5] => [5,6,4,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => ? = 2
[2,3,1,5,4,6] => [6,4,5,1,3,2] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [4,4,3,3]
 => ? = 2
[2,3,1,6,5,4] => [4,5,6,1,3,2] => ([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2,2]
 => ? = 3
[2,3,6,1,5,4] => [4,5,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,2,2]
 => ? = 3
[2,4,1,3,6,5] => [5,6,3,1,4,2] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,3,3,3,3]
 => ? = 6
[2,4,1,5,3,6] => [6,3,5,1,4,2] => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [4,3,3,3]
 => ? = 1
[2,4,1,6,3,5] => [5,3,6,1,4,2] => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => ? = 4
[2,4,1,6,5,3] => [3,5,6,1,4,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,2,2,2,2]
 => ? = 2
[2,4,3,1,6,5] => [5,6,1,3,4,2] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => [3,3,3,3,2,2]
 => ? = 4
[2,4,3,6,1,5] => [5,1,6,3,4,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,2]
 => ? = 4
[2,4,3,6,5,1] => [1,5,6,3,4,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => [3,3,3,3,1]
 => ? = 4
[2,5,1,3,6,4] => [4,6,3,1,5,2] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => [3,3,3,3,3]
 => ? = 5
Description
The multiplicity of the largest part of an integer partition.